Keywords: genetic algorithm, multi-objective optimization, truss, non-linear, line search, arc-length.

Optimization and structural analysis models are two main elements in the design optimisation of truss structures. Therefore, the degree of optimality in the design of pin-connected trusses by linear response is varied with the use of a realistic non-linear response. In this paper, the variation in the design optimisation by linear and non-linear response is considered with multi-objective genetic algorithm optimization method with the multi-populations. While, in optimization model, pareto ranking based on the multi-objective genetic algorithm method is employed, line-search with arc-length method (cylindrical arc-length technique) is used for the non-linear response in the structural analysis model. Considering that one of the preferable approaches used by both linear and non-linear analysis is that the total potential energy has a stationary value at the equilibrium position, the objective functions minimize the weight of the pin-connected truss together with its strain energy that exists in structural mechanic.

In the following part, the proposed design model is presented by introducing: firstly the optimisation, then the structural analysis model with a comparison of the other strategies.

In the design of pin-connected trusses, genetic algorithms have received considerable attention regarding their potential as a novel approach to multi-objective optimization approach. Therefore, various approaches based on the fitness assignment are adapted for the implementation with the genetic algorithm. These approaches can roughly be classified as: a) vector evaluation, b) weighted-sum, c) pareto based, d) compromise and e) goal programming [1]. The method of pareto-ranking which is managed by fitness assignment was first suggested by Goldberg [2], as a means of achieving equal reproductive potential for all pareto individuals. In this method, the population is ranked on the basis of non-dominated individuals. However, the fitness sharing must be implemented for maintaining the diversity as a result of genetic drift. After the sharing process, an appropriate selection method is used to reproduce a new generation.

As to the structural analysis model, the geometric non-linearity due to either the strain or the displacement, or both, response causes the equilibrium equations to be non-linear. Most of the early work on the solution of the non-linear equations are primarily based on incremental procedures. In later works, an alternative approach to the incremental procedure, the iterative procedure (Newton or Newton Raphson) is also used. However, due to uncertainty about the obtained maximization in load-deformation response, iterative failure does not corresponded to the actual structural failure. This issue leads the development new solution methods based on load or displacement control. One class of these methods is continuation methods. The working principle of the continuation methods is based on tracing the equilibrium path by accounting for the limit points. Some of the continuum methods are quasi-newton, line-search and arc-lengths [3]. Quasi-newton methods are developed in order to speed up the convergence of modified-Newton method, due to existing singular points. The line-search procedure is more successful in obtaining the iterative direction in an inexpensive way and thus, may be viewed as a useful tool to increase the robustness of the Newton methods. As to arc-length methods, they are developed as a remedy in cases when the solutions exhibit limit load states, and work by following a prescribed path around the previous converged equilibrium state. The compatibility in the working of the principle of the arc-length method leads to it being hybridized with the line-search method.

In order to demonstrate the variation in the optimal design with regard to the linear and non-linear response, two of well-known optimization examples are designed. As a result, it is shown that the effect of nonlinearity causes the increase in the weight of the truss structure.

1

C. Fonseca, P. Fleming, "An overview of evolutionary algorithms in multi objective optimization", Evolutionary Computation, 3(1), 1-16, 1995. doi:10.1162/evco.1995.3.1.1

2

D. Goldberg, "Genetic algorithms in search, optimization machine learning", Addson-Wesley, Reading, MA, 1989.

3

M.A. Crisfiled, "Non-linear Finite Element Analysis of Solids and Structures Vol 1-2", J. Wiley & Sons, New York, 1991 and 1997.

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